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Change from a general equation of a plane to the normal one

−−→
point M would be, the projection of its radius vector OM(x, y, z) on the normal is constant and
−−→ −−→
− →0
equals to p : pr−→ OM = p or OM · n = p or in the view of coordinates:
B 0
x cos α + y cos β + z cos γ − p = 0. (8.7)

An equation (8.7) is called the normal equation of the plane.

Remark 8.2. For the equation to be normal two following conditions must be
met:
2
2
The sum of coefficients near variables x, y and z equal to one, as cos α + cos β +
2
cos γ = 1.
Minus has to be put before the free term, since p ≥ 0 — is the length of the segment
ON or the distance from the beginning of coordinates to the plane.

8.3. Change from a general equation of a plane to the normal one

Let’s show how it is possible to reduce the general equation of the plane to the normal view.
Suppose we have got the equation Ax + By + Cz + D = 0. Let’s multiply this equation by
some digit µ, then: µAx + µBy + µCz + µD = 0. Let’s assume the following conditions for
the normal equation are satisfied by:
2
2
2
1. (µA) + (µB) + (µC) = 1;
2. µD < 0.
From the first condition we will obtain µ :
1
µ = ±√ , (8.8)
A + B + C 2
2
2
where the sign is chosen according to the second condition, and has to be opposite to the sign
of the free term D in the general equation. In case D = 0 the sign of µ is chosen sporadically.
Digit µ, obtained according to the formula (8.8), is called the normal multiplier.

8.4. Distance from the point to a plane

Suppose we have plane (p) in the system of coordinates Oxyz. That plane is given by means
−→
of normal n with the direction from the beginning of coordinates, that creates angles α, β, γ
with corresponding axes of coordinates. Let’s suppose the length of the segment determining
the distance from the beginning of coordinates to the plane, equals to d (fig. 8.7). And let’s have
point M 0 (x 0 , y 0 , z 0 ), which doesn’t belong to the plane.
Let’s resolve the task to find the distance from point M to the plane. Let’s choose an orth
−→0
vector of the normal one: n = (cos α, cos β, cos γ) and write a radius vector of point M .
Then the projection of this vector into a normal can be determined this way:

−−→
pr−→ OM = p ± d
B 0
−−→ −→0
or OM· n = p±d. In these formulas the plus sign will be only when point M and the beginning
of coordinates are located on different sides from the plane and the minus sign in vice versa. Let’s
write down the last equation in the form of coordinates:x 0 cos α+y 0 cos β +z 0 cos γ −p = ±d.
Length d is positive. The expression in the left part as the result of substitution of point’s M

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